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9 Regression with Time Series

9.1 Some Basic Concepts

Static Models

(1) yt = 0 + 1 x t + u t

t = 1 , 2 , . . . , T , where T is the number of ob-servation in the time series. The relation be-tween y and x is contemporaneous.

Example: Static Phillips Curve:

ination t = 0 + 1 unemployment t + u t .

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Finite Distributed Lag Model (FDL)

In FDL models earlier values of one or moreexplanatory variables affect the current valueof y.

(2) yt = 0 + 0 x t + 1 x t 1 + 2 x t 2 + u t

is a FDL of order two .

Multipliers

Multipliers indicate the impact of a unit changein x on y.

Impact Multiplier: Indicates the immediateone unit change in x on y. In (2) 0 is theimpact multiplier.

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To see this, suppose xt is constant, say c,

before time point t, increases by one unit toc + 1 at time point t and returns back to cat t + 1 . That is

, x t 2 = c, x t 1 = c, x t = c + 1 , x t+1 = c, x t+2 = c, . . .

Suppose for the sake of simplicity that theerror term is zero, then

yt 1 = 0 + 0 c + 1 c + 2 c

yt = 0 + 0 ( c + 1 ) + 1 c + 2 cyt+1 = 0 + 0 c + 1 ( c + 1 ) + 2 c

yt+2 = 0 + 0 c + 1 c + 2 ( c + 1)

yt+3 = 0 + 0 c + 1 c + 2 c

from which we ndyt yt 1 = 0 ,

which is the immediate change in yt .

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In the next period, t + 1 , the change is

yt+1 yt 1 = 1 ,

after that

yt+2 yt 1 = 2 ,

after which the series returns to its initiallevel yt+3 = yt 1 . The series {0 , 1 , 2 } iscalled the lag distribution, which summarizesthe dynamic effect that a temporary increasein x has on y.

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Lag Distribution: A graph of j as a func-

tion of j . Summarizes the distribution of theeffects of a one unit change in x on y as afunction of j , j = 0 , 1 , . . . .

Particularly, if we standardize the initial value

of y at yt 1 = 0 , the lag distribution tracesout the subsequent values of y due to a one-unit, temporary change in x.

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Interim multiplier of order J :

(3) ( J ) =J

j =0 j .

Indicates the cumulative effect up to J of aunit change in x on y. In (2) e.g., (1) = 0 + 1 .

Total Multiplier: (Long-Run Multiplier)

Indicates the total (long-run) change in y asa response of a unit change in x.

(4) = j =0

j .

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Example 9.1: Suppose that in annual data

int t = 1 .6 + 0 .48 inf t 0 .15 inf t 1 + 0 .32 inf t 2 + u t ,

where int is an interest rate and inf is ination rate.

Impact and long-run multipliers?

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Assumptions

Regarding the Classical Assumption, we needto account for the dependencies in time di-mension.

Assumption (4) E [u i | x i] = 0 is replaced by(5) E [u t | x s ] = 0 for all t, s = 1 , . . . , T .

In such a case we say that x is strictly exo-genous. Explanatory variables that are strictlyexogeneous cannot react to what happenedin the past. The assumption of strict exo-geneity implies unbiasedness of OLS-estimates.

A weaker assumption is contemporaneous exo-geneity:

(6) E [u t | x t ] = 0

It implies only consistency of OLS-estimates.

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Assumption

(7) C ov[ u s , u t ] = 0 , for all s = t

is the assumption of no serial correlation orno autocorrelation.

Lack of serial correlation is required for thestandard errors and the usual t- and F-statisticsto be valid.

Unfortunately, this assumption is often vio-lated in economic time series.

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9.2 Trends and Seasonality

Economic time series have a common ten-dency of growing over time. Some seriescontain a time trend.

Usually two or more series are trending overtime for reasons related to some unobservedcommon factors. As a consequence correla-tion between the series may be for the mostpart of the trend.

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Linear time trend:

(8) yt = 0 + 1 t + et .

(9) E [yt ] = 0 + 1 t .

1 > 0, upward trend,

1 < 0, downward trend.

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Exponential trend:

If the growth rate y/y of an economy is 1 .That is

(10) dy( t ) /dt

y( t ) = 1 ,

then

(11) y( t ) = y(0) e1 t .

Thus a constant growth rate leads to expo-nential trend model (c.f. continuously com-

pounded interest rate).

Typical such series are GDP, Manufacturingproduction, and CPI.

Exponential trend is modeled in practice as

(12) log( yt ) = 0 + 1 t + et ,

t = 1 , 2 , . . . , where 1 is the growth rate.

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Trending Variables in Regression

Common growth over time of series in theregression model may cause spurious regres-sion relationships. Adding a time trend to theregression eliminates usually this problem.

Example 9.3: Housing Investment and Prices:

Regressing anual observations on housing investmentper capita invpc on a housing price index price in con-stant elasticity form yields

(13a) log(invpc) = 0 .550+2 .241 log(price)

The standard error on the slope coefficient of log(price)is 0.382 , so it is statistically signicant. However,both invpc and price have upward trends. Adding atime trend yields(13b)

log(invpc) = 0 .913 0 .381 log(price) + 0 .0098 t

with a standard error of 0.679 on the price elasticityand 0.0035 on time. The time trend is statistically

signicant and it implies an approximate 1% increase

in invpc per year. The estimated price elasticity is

now negative and not statistically different from zero.

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Example 9.4: Puerto Rican Employment and the U.S.

Minimum Wage.prepop = Puerto Rican employment/popul ratio

mincov = (average minimum wage/average wage)*avgcov,

where avgcov is the proportion of workers covered by

the minimum wage law.

mincov measures the importance of minimum wagerelative to average wage.

Sample period 19501987

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The specied model is

(14a)log(prepop t ) = 0 + i log(mincov t ) + 2 log(usgnp t ) + u t .

Dependent Variable: LOG(PREPOP)Method: Least SquaresSample: 1950 1987Included observations: 38

===========================================================Variable Coefficient Std. Error t-Statistic Prob.-----------------------------------------------------------C -1.054 0.765 -1.378 0.177LOG(MINCOV) -0.154 0.065 -2.380 0.023LOG(USGNP) -0.012 0.089 -0.138 0.891===========================================================R-squared 0.660 Mean dependent var -0.944Adjusted R-squared 0.641 S.D. dependent var 0.093S.E. of regression 0.056 Akaike info criterion -2.862Sum squared resid 0.109 Schwarz criterion -2.733Log likelihood 57.376 F-statistic 34.043Durbin-Watson stat 0.340 Prob(F-statistic) 0.000===========================================================

Elasticity estimate of mincov is 0 .154 and is statisti-

cally signicant. This suggests that a higher minimumwage lowers the employment rate (as expected). The

US GNP is not statistically signicant.

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Example 9.4: Puerto Rican Employment:

Adding a trend to (14a):(14b)

log(prepop t ) = 0 + i log(mincov t )

+ 2 log(usgnp t ) + 3 t + u tproduces estimation results:

Dependent Variable: LOG(PREPOP)

Method: Least SquaresSample: 1950 1987Included observations: 38===========================================================Variable Coefficient Std. Error t-Statistic Prob.-----------------------------------------------------------C -8.729 1.300 -6.712 0.0000LOG(MINCOV) -0.169 0.044 -3.813 0.0006LOG(USGNP) 1.057 0.177 5.986 0.0000

@TREND -0.032 0.005 -6.442 0.0000===========================================================R-squared 0.847 Mean dependent var -0.944Adjusted R-squared 0.834 S.D. dependent var 0.093S.E. of regression 0.038 Akaike info criterion -3.607Sum squared resid 0.049 Schwarz criterion -3.435Log likelihood 72.532 F-statistic 62.784Durbin-Watson stat 0.908 Prob(F-statistic) 0.000===========================================================

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Seasonality

Monthly or quarterly series include often sea-sonality which shows up as regular cycles inthe series.

A common way to account for the season-ality is to include a set of seasonal dummy variables into the model.

For example, monthly data:

(15)yt = 0 + 1 feb t + 2 mar t + + 11 dec t+ 1 x t1 + + kx tk + u t

feb t , . . . , dec t are dummy variables. Januaryis the base month.

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9.3 Time Series Models

Stationarity:

A stochastic process {yt : t = 1 , 2 , . . . } is (co-variance) stationary, if

(i) E [yt ] = for all t(ii) V ar[ yt ] = 2 < for all t(iii) C ov[ yt , yt+ h ] = h for all t, i.e., the co-variance depends only on the lag length h,not time.

Example:A process with a time trend is not stationary,because its mean changes through time.

Establishing stationarity can be very difficult.However, we often must assume it since noth-ing can be learnt from time series regressionswhen the relationship between yt and xt is al-lowed to change arbitrarily out of sample.

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Weakly Dependent Series

yt is weakly dependent if yt and yt+ h are al-most independent as h .

Covariance stationary sequences are said to

be asymptotically uncorrelated if C ov[ x t , x t+ h ] 0 as h . (Intuitive char-acterization of weak dependence.)

The weak dependence replaces the notion of

random sampling implying law of large num-bers (LLN) and the central limit theorem(CLT) holds.

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Basic Time Series Processes

White Noise: Series yt is (weak) white noise(WN) if

(i) E [yt ] = for all t(ii) V ar[ yt ] = 2 < for all t(iii) C ov[ ys , yt ] = 0 for all s = t.

Remark 9.1: (i) Usually = 0 . (ii) WN-process is

stationary.

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Random Walk (RW): yt is a random walk

process if

(13) yt = yt 1 + et ,

where et WN(0 , 2e ) .

If yt RW and assuming y0 = 0 , it can beeasily shown E [yt ] = 0 ,

(14) V ar[ yt ] = t 2e ,

and

(15) C orr[ yt , yt+ h ] = tt + h .Remark 9.2: RW is a nonstationary process.

Random walk with drift:

(16) yt = + yt 1 + et , e t WN(0 , 2e ) .

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9.4 Serial Correlation and Heteroscedasticity

in Time Series Regression

Assumption 3 C ov[ u t , u t + h ] = 0 is violated if the error terms are correlated.

This problem is called the autocorrelationproblem.

Consequences of error term autocorrelationin OLS:

(i) OLS is no more BLUE(ii) Standard errors are (downwards) biased( t -statistics etc. become invalid), and thesituation does not improve for n .

However(iii) OLS estimators are still unbiased for strictlyexogeneous regressors.(iv) OLS estimators are still consistent forstationary, weekly dependent data with con-temporaneously exogeneous regressors.

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Testing for Serial Correlation

(22) yt = 0 + 1 x t1 + kx tk + u t .AR(1) errors

(23) ut = u t1 + et , et WN(0 , 2e ) .Typical procedure to test the rst order au-tocorrelation is to obtain OLS residuals u tby estimating (22), t AR(1) in the u t se-ries, and use the resulting t-statistic to inferto test H 0 : = 0 .

An alternative is to use the traditional Durbin-Watson (DW) test.

(24) DW =T

t=2( u t

u

t1 ) 2

T t=1 u

2t

.

It can be shown that

(25) DW 2(1 ) .

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Ljung-Box test

A general test for serial correlation is theLjung-Box Q-statistic,

(26) Q = T ( T + 2)k

j =1

2 j( T

j )

.

If the null hypothesis

(27) H 0 : 1 = 2 = = k = 0is true Q has the asymptotic 2k distribution.

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Serial Correlation-Robust Inference with OLS

The idea is to nd robust standard errors forthe OLS estimates.

Again using matrix notations simplies con-

siderably exposure. Let

(28) y = X + u ,

where

(29) C ov[ u ] = u ,

Again, write

(30) = + ( X X ) 1 X u .Then

(31) C ov[ ] = ( X X ) 1 ( X X ) 1 .The problem is how to estimate = X u X .

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Newey and West (1987)suggest and estima-

tor(32) = T T k

T t =1 u

2t x t x t

+ qv=1 1 vq+1

T t = v+1 ( x t u t u t v x t v + x t v u t v u t x t )

which is supposed to be robust both against

heteroscedasticity and autocorrelation. Theq-variable is determined as a function of thenumber of observations.

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Example 9.5: Puerto Rican Wage:

Correlogram of Residuals

Sample: 1950 1987Included observations: 38

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 0.443 0.443 8.0658 0.0052 0.153 -0.054 9.0537 0.011

3 0.085 0.047 9.3684 0.0254 0.065 0.020 9.5595 0.0495 -0.143 -0.228 10.500 0.0626 -0.167 -0.020 11.821 0.0667 -0.243 -0.189 14.707 0.0408 -0.270 -0.115 18.397 0.0189 -0.288 -0.120 22.740 0.007

10 -0.081 0.112 23.098 0.01011 -0.117 -0.153 23.865 0.01312 -0.063 0.019 24.099 0.02013 -0.008 -0.035 24.104 0.03014 -0.035 -0.188 24.179 0.04415 -0.070 -0.050 24.503 0.05716 0.068 0.027 24.819 0.073

The residuals are obviously autocorrelated. Using the

the above autocorrelation robust standard errors yields

the following results.

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Dependent Variable: LOG(PREPOP)

Method: Least SquaresSample: 1950 1987Included observations: 38Newey-West HAC Standard Errors & Covariance (lag truncation=3)==========================================================Variable Coefficient Std. Error t-Statistic Prob.----------------------------------------------------------C -8.728657 1.589053 -5.492994 0.0000LOG(MINCOV) -0.168695 0.038469 -4.385230 0.0001LOG(USGNP) 1.057351 0.219516 4.816736 0.0000@TREND -0.032354 0.006605 -4.898514 0.0000==========================================================R-squared 0.847089 Mean dep. var -0.944074Adjusted R-squared 0.833597 S.D. dep. var 0.092978S.E. of regression 0.037928 Akaike info crit. -3.606957Sum squared resid 0.048910 Schwarz criterion -3.434580Log likelihood 72.53218 F-statistic 62.78374Durbin-Watson stat 0.907538 Prob(F-statistic) 0.000000==========================================================

We observe that particularly the standard error of

log( usgnb ) increases compared to Example 9.4.

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Estimating the residual autocorrelation as an

AR-process

An alternative to the robustifying of the stan-dard errors with the Newy-White procedure(32), is to explicitly model the error term as

an autoregressive process as is done in equa-tion (23) and estimate it.

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Example 9.6: Estimation the Puerto Rican example

with AR(1) errors yields:

=================================================================Dependent Variable: LOG(PREPOP)Method: Least SquaresSample (adjusted): 1951 1987Included observations: 37 after adjustmentsConvergence achieved after 14 iterations=================================================================

Variable Coefficient Std. Error t-Statistic Prob.-----------------------------------------------------------------C -5.256865 1.492220 -3.522850 0.0013LOG(MINCOV) -0.090233 0.048103 -1.875824 0.0698LOG(USGNP) 0.588807 0.207160 2.842278 0.0077@TREND -0.019385 0.006549 -2.959877 0.0058AR(1) 0.701498 0.123959 5.659112 0.0000=================================================================R-squared 0.907075 Mean dependent var -0.949183

Adjusted R-squared 0.895459 S.D. dependent var 0.088687S.E. of regression 0.028675 Akaike info criterion -4.140503Sum squared resid 0.026312 Schwarz criterion -3.922811Log likelihood 81.59930 F-statistic 78.09082Durbin-Watson stat 1.468632 Prob(F-statistic) 0.000000=================================================================

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Residual autocorrelations:

Correlogram of Residuals

Sample: 1951 1987Included observations: 37Q-statistic probabilities adjusted for 1 ARMA term(s)

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 0.187 0.187 1.40712 0.078 0.044 1.6551 0.1983 0.097 0.078 2.0557 0.3584 0.085 0.053 2.3723 0.4995 -0.273 -0.320 5.7274 0.2206 -0.016 0.088 5.7395 0.3327 -0.013 -0.005 5.7475 0.4528 -0.122 -0.096 6.4908 0.4849 -0.226 -0.156 9.1276 0.332

10 0.061 0.064 9.3267 0.408

The results indicate no additional autocorrelation in

the residuals.

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Residual Autocorrelation and Common Fac-

tor

The lag operator L is dened as Ly t = yt 1(generally L pyt = yt p).

Thus, equation (23) can be written as

(33) (1 L ) u t = et

or

(34) u t = et

1 LUsing this in (22), we can write (for the sim-plicity, assume k = 1 and denote x t = x t 1 )

(35) yt = 0 + 1 x t + et

1 Lor(36)(1 L ) yt = (1 L ) 0 + 1 (1 L ) x t + e t

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This implies that the dynamics of yt and xtshare 1 L in common, called common fac-tor.

Thus, the autocorrelation in ut is equivalentthat there is a common factor in the regres-

sion in (23).

This can be tested by estimating the unre-stricted regression

(37) yt = 0 + 1 yt

1 + 2 x t + 3 x t

1 + et

and testing whether it satises restrictionsimplied by (36), which can be written as(38)yt = (1 ) 0 + yt1 + 1 x t 1 x t1 + et

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That is, whether

(39) 3 = 1 2If this hypothesis is not accepted, the ques-tion is of wrong dynamic specication of themodel, not autocorrelation in the residuals.

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Example 9.7: Puerto Rican example. The unrestricted

model estimates are:

Dependent Variable: LOG(PREPOP)Method: Least SquaresSample (adjusted): 1951 1987Included observations: 37 after adjustments===========================================================Variable Coefficient Std. Error t-Statistic Prob.-----------------------------------------------------------

C -5.612596 1.547354 -3.627223 0.0011LOG(PREPOP(-1)) 0.535366 0.124932 4.285246 0.0002LOG(MINCOV) -0.142230 0.047224 -3.011794 0.0052LOG(MINCOV(-1)) 0.033409 0.046831 0.713386 0.4811LOG(USGNP) 0.561893 0.188654 2.978437 0.0057LOG(USGNP(-1)) 0.137353 0.226785 0.605651 0.5493@TREND -0.019768 0.005975 -3.308752 0.0024===========================================================R-squared 0.928815 Mean dependent var -0.949183

Adj R-squared 0.914578 S.D. dependent var 0.088687S.E. of reg 0.025921 Akaike info criterion -4.298904Sum squared resid 0.020156 Schwarz criterion -3.994136Log likelihood 86.52972 Hannan-Quinn criter. -4.191459F-statistic 65.23934 Durbin-Watson stat 1.634219Prob(F-statistic) 0.000000===========================================================

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Testing for the implied restriction by AR(1)-residuals

by imposing restrictions in EViews ( View > CoefficientTests > Wald-Coefficient Restrictions . . . )

gives:

Wald Test:Equation: EX97=================================================Test Statistic Value df Probability-------------------------------------------------F-statistic 4.436898 (2, 30) 0.0205Chi-square 8.873795 2 0.0118=================================================

Null Hypothesis Summary:=================================================

Normalized Restriction (= 0) Value Std. Err.-------------------------------------------------C(2)*C(3) + C(4) -0.042736 0.033279C(2)*C(5) + C(6) 0.438171 0.150355=================================================

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The null hypothesis is rejected, which implies that

rather than the error term is autocorrelated the modelshould be specied as

yt = 0 + 1 yt1 + 1 x t1 + 11 x t1 ,1 + 2 x t, 2 + 21 x t1 ,2 + et ,

where y = log( prepop ) , x1 = log( mincov ) , x2 = log( usgnp ) ,

and x3 = trend .

Finally, because the coefficient estimates of log( mincov t1 )and log( usgnp t1 ) are not statistically signicant, theycan be dropped from the model, such that the nal

model becomes:

Dependent Variable: LOG(PREPOP)Method: Least SquaresIncluded observations: 37 after adjustments==========================================================Variable Coefficient Std. Error t-Statistic Prob.----------------------------------------------------------C -5.104888 1.188371 -4.295701 0.0002LOG(PREPOP(-1)) 0.558090 0.103817 5.375713 0.0000LOG(MINCOV) -0.106787 0.031282 -3.413647 0.0018LOG(USGNP) 0.630332 0.154822 4.071322 0.0003

@TREND -0.017463 0.004849 -3.601696 0.0011==========================================================R-squared 0.926226 Mean dependent var -0.949183Adj R-squared 0.917004 S.D. dependent var 0.088687S.E. of reg 0.025550 Akaike info criter. -4.371286Sum sqerd resid 0.020889 Schwarz criterion -4.153594Log likelihood 85.86878 Hannan-Quinn criter. -4.294539F-statistic 100.4388 Durbin-Watson stat 1.468436Prob(F-statistic) 0.000000==========================================================

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Autocorrelation has also disappeared from the residu-

als:

Sample: 1951 1987Included observations: 37==============================================================Autocorr Partial Corr AC PAC Q-Stat Prob--------------------------------------------------------------

. |*. | . |*. | 1 0.203 0.203 1.6465 0.199

. |*. | . | . | 2 0.079 0.039 1.9008 0.387

. |*. | . | . | 3 0.092 0.072 2.2586 0.521

. | . | . | . | 4 0.018 -0.017 2.2724 0.686***| . | ***| . | 5 -0.346 -0.372 7.6823 0.175

.*| . | . | . | 6 -0.111 0.020 8.2560 0.220

.*| . | . | . | 7 -0.067 -0.010 8.4747 0.293

.*| . | . | . | 8 -0.124 -0.052 9.2354 0.323

.*| . | .*| . | 9 -0.143 -0.091 10.2920 0.327

. |*. | . |*. | 10 0.156 0.110 11.5860 0.314==============================================================

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Autoregressive Conditional Heteroscedastic-

ity ARCH

Temporal volatility clustering is typical forspeculative series like stock returns, interestrates, currencies, etc.

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Example 9.7: The following time series plot shows Mi-

crosofts weekly (log) returns ( r t = 100 log( P t /P t1 ) )for the sample period from January 1990 through Oc-

tober 2006.

-30

-20

-10

0

10

20

30

90 92 94 96 98 00 02 04 06

R e

t u r n

Week

Microsoft Weekly Returns [1990 - 2006]

Figure 9.1: Microsoft weekly returns for the sample

period January 1990 through November 2006.

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-12

-8

-4

0

4

8

90 92 94 96 98 00 02 04 06

SP500 Weekly Returns [1990 - 2006]

R e

t u r n

Week

Figure 9.1: SP500 weekly returns for the sample pe-

riod January 1990 through November 2006.

There is probably some volatility clustering present in

both return series.

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Engle (1982) suggested modeling the clus-

tering volatility with the ARCH models.

A time series ut follows an ARCH(1)-process(AutoRegressive Conditional Heteroscedastic-ity) if

(40) E [u t ] = 0 ,

(41) C ov[ u t , u s ] = 0 , for all t = s,

and

(42) ht = V ar[ u t |u t1 ] = 0 + 1 u2t1 ,

where 0 > 0 and 0 1 < 1.

We say that ut follows and ARCH(1)-processand denote ut ARCH(1) .

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An important property of an ARCH process

is that if ut ARCH , then

(43) zt = ut h t WN(0,1) .

That is the standardized variable zt is whitenoise ( E [zt ] = 0 , V ar[ zt ] = 1 , and C ov[ zt , z t+ k] = 0

for all k = 0 ).

Furthermore

(44)V ar[ zt |u t1 ] = V ar

ut h t1 |u t1

= 1

h t V ar[ u t |u t1 ] = 1 .That is, zt has constant conditional variance.

This implies that there should not remain anyvolatility clustering in zt .

This can be checked by investigating the au-tocorrelations of the squared zt -series, z2t , forexample with the Ljung-Box Q-statistic, de-ned in (26).

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It may be noted further that if zt N (0 , 1) ,

then

(45) ut |u t1 N (0 , h t ) .

Remark 9.4: If (43) holds, using (43) we can alwayswrite

(46) ut = h t zt ,where zt are independent N (0 , 1) random variables.

The parameters of the ARCH-process are es-timated by the method of maximum likeli-hood (ML).

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Example 9.8: Consider the SP500 returns. With EViews

we can estimate by selecting Quick > Estimate Equation. . . , specifying rm c in the model box, selecting Esti-

mation Method:

ARCH-Autoregressive Conditional Heteroscedasticity ,

selecting ARCH option equal to 1 and GARCH option

equal to 0.Note that specifying in the model box rm c implies

that we estimate ut = rm,t E [r mt ].These specications yield the following results:

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=================================================================

Dependent Variable: RMMethod: ML - ARCH (Marquardt) - Normal distributionSample (adjusted): 1/08/1990 10/30/2006Included observations: 877 after adjustmentsConvergence achieved after 9 iterationsVariance backcast: ONGARCH = C(2) + C(3)*RESID(-1) 2=================================================================

Coefficient Std. Error z-Statistic Prob.-----------------------------------------------------------------C 0.189986 0.062065 3.061069 0.0022=================================================================

Variance Equation=================================================================C 3.237585 0.135488 23.89579 0.0000RESID(-1)^2 0.247032 0.039617 6.235534 0.0000=================================================================R-squared -0.000294 Mean dependent var 0.154412Adjusted R-squared -0.002583 S.D. dependent var 2.077382

S.E. of regression 2.080063 Akaike info criterion 4.240795Sum squared resid 3781.503 Schwarz criterion 4.257134Log likelihood -1856.588 Durbin-Watson stat 2.149700=================================================================

Thus the estimated ARCH(1) model is

(47) h t = 3 .238+0 .247 u 2t1 ,

(0 .135) (0 .0396)where ut = rm,t with = E [r m,t ], estimated fromthe sample period as = 0 .18986 (i.e., the average

weekly return has been approximately 0 .19%, or 9 .9% per year).

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Check next the autocorrelation of the squared stan-

dardized residuals zt = u t / h t .Correlogram of Standardized Residuals Squared

Sample: 1/02/1990 10/24/2006Included observations: 877

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 -0.006 -0.006 0.0274 0.869

2 0.028 0.028 0.7023 0.7043 0.098 0.098 9.1175 0.0284 0.083 0.084 15.178 0.0045 0.044 0.041 16.884 0.0056 0.104 0.093 26.502 0.0007 0.096 0.084 34.603 0.0008 0.037 0.023 35.787 0.0009 0.036 0.010 36.945 0.000

10 0.065 0.033 40.685 0.000

It seems that there is still left some volatility clustering

especially due to the longer lags. The p-value from the

third order forwards is statistically signicant. Thus

the simple ARCH(1) does not seem to fully capture

the volatility clustering in the series. We can try to

improve the model.

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GARCH

An important extension to the ARCH modelis due to Bollerslev (1986), which is calledGARCH, Generalized ARCH. GARCH(1,1) isof the form

(48) ht = 0 + 1 u2t1 + h t1 ,where it is assumed that 0 > 0 and0 < + < 1.

Remark 9.5: If > 0 then must also be > 0.

The GARCH-term ht1 essentially accumu-lates the historical volatility and indicatesthe persistence of the volatility.

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Example 9.9: GARCH(1,1) model for the SP500 re-

turns

Dependent Variable: RMMethod: ML - ARCH (Marquardt) - Normal distributionSample (adjusted): 1/02/1990 10/24/2006Included observations: 877 after adjustmentsConvergence achieved after 17 iterationsVariance backcast: ONGARCH = C(2) + C(3)*RESID(-1) 2 + C(4)*GARCH(-1)

==============================================================Coefficient Std. Error z-Statistic Prob.

--------------------------------------------------------------C 0.197385 0.058686 3.363410 0.0008==============================================================

Variance Equation==============================================================C 0.023269 0.013163 1.767719 0.0771RESID(-1)^2 0.057345 0.012168 4.712861 0.0000

GARCH(-1) 0.937023 0.012920 72.52709 0.0000==============================================================R-squared -0.000428 Mean dependent var 0.154412Adjusted R-squared -0.003866 S.D. dependent var 2.077382S.E. of regression 2.081394 Akaike info criterion 4.119464Sum squared resid 3782.013 Schwarz criterion 4.141250Log likelihood -1802.385 Durbin-Watson stat 2.149410==============================================================

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The autocorrelations of the squared standardized resid-

uals indicate still some possible rst order autocorrela-tion remained in the series. Nevertheless the estimate

of the rst order autocorrelation, though statistically

signicant ( p-value 0.022) at the 5% level, is small by

magnitude (0 .078). Thus, we can conclude that the

GARCH(1,1) ts pretty well to the data.Correlogram of Standardized Residuals Squared

Sample: 1/02/1990 10/24/2006Included observations: 877

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 0.078 0.078 5.3952 0.0202 -0.007 -0.014 5.4446 0.0663 0.041 0.043 6.9264 0.074

4 -0.007 -0.014 6.9687 0.1385 -0.017 -0.015 7.2345 0.2046 -0.011 -0.011 7.3509 0.2907 -0.026 -0.024 7.9459 0.3378 -0.040 -0.035 9.3347 0.3159 -0.017 -0.011 9.5916 0.385

10 -0.009 -0.006 9.6665 0.470

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There are several other extensions of the ba-

sic ARCH-model (see EViews manual or help,or for a comprehensive presentation Taylor,Stephen J. (2005). Asset Price Dynamics,Volatility, and Prediction . Princeton Univer-sity Press).

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ARCH in regression

Consider the simple regression

(49) yt = 0 + 1 x t + u t .

If the error term ut follows a GARCH-process

then accounting for it and estimating the re-gression parameters with the method of max-imum likelihood rather than the OLS yieldmore accurate estimates.

Particularly, if the ARCH-effect is strong, OLSmay lead highly unstable estimates to 0 and1 (usually, however, OLS works pretty well).

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Example 9.11: Consider next the market model of

Microsoft weekly returns on SP500.Estimating the market model

(50) r t = 0 + 1 r m + u t ,

with OLS yields

Estimation results:

===================================================================Dependent Variable: RMethod: Least SquaresSample (adjusted): 1/08/1990 10/30/2006Included observations: 877 after adjustments===================================================================Variable Coefficient Std. Error t-Statistic Prob.-------------------------------------------------------------------C 0.272492 0.128328 2.123402 0.0340RM 1.169974 0.061639 18.98110 0.0000===================================================================R-squared 0.291660 Mean dependent var 0.453150Adjusted R-squared 0.290850 S.D. dependent var 4.500432S.E. of regression 3.789860 Akaike info criterion 5.504813Sum squared resid 12567.66 Schwarz criterion 5.515706Log likelihood -2411.861 F-statistic 360.2821Durbin-Watson stat 2.013695 Prob(F-statistic) 0.000000===================================================================

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ML with GARCH(1,1) error specication yields the

following results:

================================================================Dependent Variable: RMethod: ML - ARCH (Marquardt) - Normal distributionSample (adjusted): 1/02/1990 10/24/2006Included observations: 877 after adjustmentsConvergence achieved after 20 iterationsVariance backcast: ON

GARCH = C(3) + C(4)*RESID(-1) 2 + C(5)*GARCH(-1)================================================================

Coefficient Std. Error z-Statistic Prob.----------------------------------------------------------------C 0.262170 0.117688 2.227670 0.0259RM 1.138309 0.068997 16.49795 0.0000================================================================

Variance Equation================================================================

C 0.153567 0.058060 2.644976 0.0082RESID(-1)^2 0.038894 0.009150 4.250488 0.0000GARCH(-1) 0.949674 0.012409 76.52862 0.0000================================================================R-squared 0.291435 Mean dependent var 0.453150Adjusted R-squared 0.288184 S.D. dependent var 4.500432S.E. of regression 3.796977 Akaike info criterion 5.415744Sum squared resid 12571.65 Schwarz criterion 5.442976Log likelihood -2369.804 F-statistic 89.66397Durbin-Watson stat 2.010866 Prob(F-statistic) 0.000000================================================================

The results show that in terms of standard errors there

are no material differences.

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End of the notes: